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Matheon Project C9: Simulation and Optimization of Semiconductor Crystal Growth from the Melt Controlled by Traveling Magnetic Fields 1 - Subpage Sublimation growth

(Project was called "Numerical simulation and control of sublimation growth of semiconductor bulk single crystals" in the second funding period until of the DFG Research Center Matheon.)

Sublimation growth was the topic of the project C9 in the period 2002-2007. The results obtained since 2007 on crystal growth from the melt can be found on the main page of the project.

Quick-Links

Research Team, Collaboration, Funding

Head Persons of Research Team
Prof. Dr. Jürgen Sprekels, Dr. Olaf Klein
(Weierstrass Institute for Applied Analysis and Stochastics (WIAS), Berlin, Germany),
Prof. Dr. Fredi Tröltzsch
(Department of Mathematics, Technical University (TU), Berlin, Germany).
Current Research Team and Topical Classification
Dr. Pierre-Etienne Druet, Dr. Olaf Klein, Prof. Dr. Jürgen Sprekels, Prof. Dr. Fredi Tröltzsch, Dr. Irwin Yousept, Vili Dhamo (associated member).
Pierre-Etienne Druet, Olaf Klein, and Jürgen Sprekels are members of the research group Thermodynamic Modeling and Analysis of Phase Transition at WIAS;
Fredi Tröltzsch, Irwin Yousept and Vili Dhamo are members of the research group Modelling - Numerics - Differential Equations, Section: Optimization in PDEs at the Department of Mathematics of TU Berlin.
At WIAS, this project belongs to the application Crystal Growth, Especially under the Influence of Magnetic Fields in the Main Application Area Phase Transitions and Multi-functional Materials.

Former members
P. Philip (now LMU Munich), J. Geiser (now HU Berlin), C. Meyer (now TU Darmstadt), A. Rösch (now University of Duisburg-Essen).
Related Projects:
In Collaboration With
Leibniz Institute of Crystal Growth (IKZ), Berlin, Germany
Funding
This project is currently supported by the Deutsche Forschungsgemeinschaft (DFG) within the DFG Research Center Matheon "Mathematics for Key Technologies".
Matheon-Logo
Until the end of 2003, the project was also supported by the German Ministry of Education and Research (BMBF) within the program "New Mathematical Methods in Manufacturing and Service Industry" #03SPM3B5. The authors are responsible for the contents of this publication. 2

Description of Sublimation Growth

Owing to numerous technical applications in electronic and optoelectronic devices, such as lasers, semiconductors and sensors, the industrial demand for high quality SiC bulk single crystals is increasing. Sublimation growth of SiC bulk single crystals via physical vapor transport (PVT), also known as the modified Lely method, has been one of the most successful and most widely used growth techniques of recent years. Usually, a PVT growth system consists of a graphite crucible containing polycrystalline SiC source powder and a single-crystalline SiC seed (see Fig. 1).

setup-small.gif  - Setup of the growth apparatus

Fig. 1: Setup of growth apparatus according to M. PONS, M. ANIKIN, K. CHOUROU, J.M. DEDULLE, R. MADAR, E. BLANQUET, A. PISCH, C. BERNARD, P. GROSSE, C. FAURE, G. BASSET, Y. GRANGE, State of the art in the modelling of SiC sublimation growth , Mater. Sci. Eng. B 61-62 (1999), 18-28.

The crucible is intensely heated e.g. by induction heating. The source powder is placed in the hot zone of the growth apparatus, whereas the seed crystal is cooled by means of a blind hole, establishing a temperature difference between source and seed. Growth temperature can reach up to 3000 K for growth of the SiC polytype 6H. The high temperature and the low pressure cause the source powder to sublimate, adding molecules made up of silicon and carbon to the gas phase. The composition of the gas mixture has been analysized both experimentally and by thermodynamical theorectical considerations, showing that, apart from the inert gas, Si, Si2C, and SiC2 constitute the predominant species. As the SiC source is kept at a higher temperature than the cooled SiC seed, sublimation is encouraged at the source and crystallization is encouraged at the seed, causing the partial pressures of Si, Si2C, and SiC2 to be higher in the neighborhood of the source and lower in the neighborhood of the seed. As the system tries to equalize the partial pressures, source material is transported to the seed which grows into the reaction chamber.

There is also a variant of PVT where tantalum is used for the crucible material instead of graphite.

Many experimental and theoretical publications of recent years have treated PVT with the goal of improving quality and size of the grown crystals while trying to increase the growth rate at the same time.

Since reactor temperatures up to 3000 K make direct observations of the processes inside the growth chamber highly impractical, it is of paramount importance to have reliable theoretical models that allow the use of numerical simulation to improve the understanding of the growth procedure and to predict favorable growth conditions.

The physical and mathematical modeling of the growth process leads to a highly nonlinear system of coupled partial differential equations. In addition to the kinetics of a rare gas mixture at high temperatures one has to consider heat transport by conduction and radiation, reactive matter transport through porous and granular media and different kinds of chemical reactions and phase transitions. Moreover, several free boundaries occur in the model.

The Simulation Software WIAS-HiTNIHS

One of the main objectives of this project is to develop the model into a simulation tool to optimize the growth process. To that end, the software WIAS-HiTNIHS (pronounce: ~hit-nice) (High Temperature Numerical Induction Heating Simulator) has been and continues to be developed. It is based on the program package pdelib and uses the grid generator Triangle, the interpreted script language Lua, the graphics software OpenGL, together with its X-Windows based emulation Mesa, and the sparse matrix solver PARDISO.

In its current version the software developed during the project constitutes an effective tool to simulate the magnetic potential distributions, heat source distributions, and both transient and stationary temperature fields inside the growth apparatus including heat transport by diffusion, radiation, and constant convection in good agreement with physical experiments. It can thus be used to test and optimize geometrical setups with respect to favorable growth conditions.

An optimization module allows the control of parameters such as heating power and coil position with the objective of minimizing functionals such as the max-norm of the radial temperature gradient in a neighborhood of the growing crystal's surface.

Please also see the WIAS-HiTNIHS Main Page.

centerTemp.gif = 
Computed Temperature Distribution
At the Leibniz Institute of Crystal Growth (IKZ), Berlin, Germany a version of WIAS-HiTNIHS was installed and used for numerical simulations aiding growth experiments.

Modeling

Gas Phase, Crystal Growth

The model of the gas mixture is based on continuous mixture theory and results in balance equations for mass, momentum and energy, including reaction-diffusion equations. The energy balance is a nonlinear parabolic PDE, whereas the mass and momentum balance equations have the form of Euler or Navier-Stokes equations, depending on the magnitude of the viscosity.

Due to phase transitions and chemical reactions, the mathematical model of the SiC sublimation growth process involves several free boundaries, namely the surfaces of the growing crystal, the sublimating source, and the reacting graphite wall.

As a first over-simplified approximation, one can consider the gas mixture as a homogeneous SiC gas, computing the mass flux from the gas to the crystal via a Hertz-Knudsen formula. However, if the model is to include the observed graphitization of the powder source, it becomes necessary to consider the different constituents of the ``SiC-gas'', consisting of Si, Si2C, SiC2, etc. Instead of simple sublimation, one has to consider several chemical reactions occurring at the surfaces. Then, instead of equilibrium pressures for the different components, one merely has relations between the different partial pressures in the equilibrium, resulting from the mass action laws of the considered chemical reactions.

Heat Transfer in Solid Components

The energy balance in the gas mixture is coupled to the different energy balance equations in the various solid components of the growth apparatus via suitable interface conditions. Due to the high temperatures, it is essential to include heat transfer by radiation between surfaces of cavities in the growth apparatus. In our model we account for radiative heat transfer using the net radiation model. The semi-transparency of the single crystal is included via the band approximation model.

The heat sources caused by induction heating are computed via an axisymmetric complex-valued magnetic scalar potential that is determined as the solution of an elliptic PDE using the imposed voltage as input data. The scalar potential allows to calculate the resulting current density and thus the heat sources.

Material Data

In collaboration with the IKZ most of the relevant material data could be determined by measurements and from the existing literature.

Mathematical Analysis

Finite volume schemes for, both transient and stationary, nonlinear integro-partial differential equations modeling the diffusive-radiative heat transfer in the system are studied mathematically. The analysis focusses on discrete existence and uniqeness results, maximum principles, as well as the convergence of the scheme.

Another analytical issue is the extension of available existence and uniqueness results for weak solutions to the system consisting of the heat equation with nonlocal radiation boundary conditions coupled to the Maxwell equations (electrotechnical approximation). In order to rigorously treat optimal induction heating problems, it is important to in particular deal with Joule heating sources: A mathematical challenge since this term of the PDEs has a quadratic growth in the derivatives of the magnetic field.

Using available or/and extended results on the existence of a continuous solution for for parabolic PDEs linear diffusion coefficients, but with nonlocal interface conditions as they arise from the modeling of diffuse-gray radiation, a mathematical theory of optimal control is developed to allow the control of parameters such as heating power and coil position with the objective of minimizing functionals such as the max-norm of the radial temperature gradient in a neighborhood of the growing crystal's surface.

Publications

The following list restricts to the publications on sublimation growth. The publications concerning the new topic of Project C9, crystal growth from the melt, are listed on the main page of the project.

Theses

  1. I. Yousept: Optimal control of partial differential equations involving pointwise state constraints: Regularization and applications. Doctoral thesis, successfully defended at the Department of Mathematics, Technical University of Berlin, Germany, 2008.

  2. C. Meyer: On a semilinear elliptic optimal control problem with nonlocal radiation boundary conditions. Doctoral thesis, successfully defended at the Department of Mathematics, Technical University of Berlin, Germany, 2006.

  3. P. Philip: Transient Numerical Simulation of Sublimation Growth of SiC Bulk Single Crystals. Modeling, Finite Volume Method, Results. Doctoral thesis, successfully defended at the Department of Mathematics, Humboldt University of Berlin, Germany, Febuary 03, 2003. Report No. 22, Weierstrass-Institut für Angewandte Analysis und Stochastik, Berlin.

    Using the following link to an overview weg-page, you can download the thesis: WIAS-Report-22.

Articles in Refereed Scientific Journals

  1. J. Geiser, O. Klein, P. Philip: A finite volume scheme designed for anisotropic heat transfer in general polyhedral domains, Adv. Math. Sci. Appl. (AMSA), 18 (2008), 43-67.

    The paper is a revised version of a preprint, which can be downloaded via the following links:
    Postscript- and pdf-versions of the WIAS preprint are available via the overview web-page WIAS-Preprint-1033. Alternatively, the preprint version is also available as a postscript or pdf file via the DFG Research Center Matheon Preprints web-page. It is found as the 256th preprint in the list on that page.
    A pdf-version of the IMA preprint is available using the following link: IMA-Preprints May 2005.

  2. A. Rösch, F. Tröltzsch: On regularity of solutions and Lagrange multipliers of optimal control problems for semilinear equations with mixed pointwise control-state constraints, SIAM Journal Control and Optimization, 46 (2007), 1098-1115.

  3. C. Meyer: Second-order sufficient optimality conditions for a semilinear optimal control problem with nonlocal radiation boundary conditions, ESAIM Optim. Cal. Var., 13 (2007), 750-775.

  4. A. Rösch, F. Tröltzsch: Sufficient second-order optimality conditions for an elliptic optimal control problem with pointwise control-state constraints, SIAM Journal Control and Optimization, 17 (2006), 776-794.

  5. A. Rösch, F. Tröltzsch: Existence of regular Lagrange multipliers for a nonlinear elliptic optimal control problem with pointwise control-state constraints, SIAM Journal Control and Optimization, 45 (2006), 548-564.

  6. C. Meyer, A. Rösch, F. Tröltzsch: Optimal Control of PDEs with regularized pointwise state constraints, Computational Optimization and Applications, 33 (2006), 209-228.

  7. A. Rösch: Error estimates for linear-quadratic control problems with control constraints, Optim. Meth. Softw., 21 (2006), 121-134.

  8. C. Meyer, P. Philip, F. Tröltzsch: Optimal Control of a Semilinear PDE with Nonlocal Radiation Interface Conditions, SIAM Journal on Control and Optimization, 45 (2006), 699-721.

    The paper is minorly revised from the preprint version, which is a available via the following four links:
    Postscript- and pdf-versions of the WIAS preprint are available via the overview web-page WIAS-Preprint-976.
    Postscript- and pdf-versions of the TU Berlin preprint are available via the overview web-page TU-Preprint-2004/33.
    Alternatively, the preprint version is also available as a postscript or pdf file via the DFG Research Center Matheon Preprints web-page. It is found as the 37th preprint in the list on that page.
    A pdf-version of the IMA preprint is available using the following link: IMA-Preprint-2002.pdf.

  9. J. Geiser, O. Klein, P. Philip: Influence of anisotropic thermal conductivity in the apparatus insulation for sublimation growth of SiC: Numerical investigation of heat transfer, Crystal Growth & Design, 6 (2006), 2021-2028.

    The paper is minorly revised with respect to its preprint version, available via the following links:
    Postscript- and pdf-versions of the WIAS preprint are available via the overview web-page WIAS-Preprint-1034. Alternatively, the preprint version is also available as a postscript or pdf file via the DFG Research Center Matheon Preprints web-page. It is found as the 257th preprint in the list on that page.
    A pdf-version of the IMA preprint is available using the following link: IMA-Preprints May 2005.

  10. J. Geiser, O. Klein, P. Philip: Transient numerical study of temperature gradients during sublimation growth of SiC: Dependence on apparatus design, Journal of Crystal Growth, 297 (2006), 20-32.

  11. Casas, M. Mateos, F. Tröltzsch: Error estimates for the numerical approximation of boundary semilinear elliptic control problems, Comput. Optim. Appl., 31 (2005), 193-219.

  12. C. Meyer, P. Philip: Optimizing the temperature profile during sublimation growth of SiC single crystals: Control of heating power, frequency, and coil position, Crystal Growth & Design, 5 (2005), 1145-1156.

    This article has been minorly revised from its preprint version which is available as a postscript or pdf file via the overview web-page WIAS-Preprint-895.
    Alternatively, the preprint version is also available as a postscript or pdf file via the DFG Research Center Matheon Preprints web-page. It is found as the 37th preprint in the list on that page.

  13. O. Klein, P. Philip: Transient conductive-radiative heat transfer: Discrete existence and uniqueness for a finite volume scheme, accepted for publication in M3AS: Mathematical Models and Methods in Applied Sciences.

    This article has been revised from its preprint version which is available as a postscript or pdf file via the overview web-page WIAS-Preprint-871.
    Alternatively, the preprint version is also available as a postscript or pdf file via the DFG Research Center Matheon Preprints web-page. It is found as the 3rd preprint in the list on that page.

  14. C. Meyer, A. Rösch: Superconvergence properties of optimal control problems, SIAM Journal Control and Optimization, 43 (2004), 970-985.

  15. O. Klein, P. Philip, J. Sprekels: Modeling and simulation of sublimation growth of SiC bulk single crystals, Interfaces and Free Boundaries, 6 (2004), 295-314.

  16. A. Rösch, F. Tröltzsch: Sufficient-second order optimality conditions for a parabolic optimal control problem with pointwise state constraints, SIAM Journal Control and Optimization, 42(1) (2003), 138-154.

  17. O. Klein, P. Philip: Transient temperature phenomena during sublimation growth of silicon carbide single crystals, Journal of Crystal Growth, 249 (2003), 514-522.

    This article has been slightly revised from its preprint version which is available as a postscript or pdf file via the overview web-page WIAS-Preprint-755.
    Alternatively, the preprint version is also available as a postscript or pdf file via the DFG Research Center Matheon Preprints web-page. It is found as the 38th preprint in the list on that page.

  18. O. Klein, P. Philip: Transient numerical investigation of induction heating during sublimation growth of silicon carbide single crystals, Journal of Crystal Growth, 247 (2003), 219-235.

    This article has been revised from its preprint version which is available as a postscript or pdf file via the overview web-page WIAS-Preprint-659.

  19. N. Arada, E. Casas, F. Tröltzsch}: Error estimates for a semilinear elliptic optimal control problem, Computational Optimization and Approximation, 23 (2002), 201-229.

  20. E. Casas, F. Tröltzsch}: Error estimates for the finite-element approximation of a semilinear elliptic control problem, Control and Cybernetics, 31 (2002), 695-712.

  21. O. Klein, P. Philip: Correct voltage distribution for axisymmetric sinusoidal modeling of induction heating with prescribed current, voltage, or power, IEEE Transactions on Magnetics, 38 (2002), no. 3, 1519-1523.

    This article has been slightly revised from its preprint version which is available as a postscript or pdf file via the overview web-page WIAS-Preprint-669.

  22. A. Rösch, F. Tröltzsch}: Sufficient second order optimality condititions for a state-constrained optimal control problem of a weakly singular integral equation, Num. Funct. Anal. Appl., 23 (2002), 173-193.

  23. O. Klein, P. Philip, J. Sprekels, K. Wilmanski: Radiation- and convection-driven transient heat transfer during sublimation growth of silicon carbide single crystals, Journal of Crystal Growth, 222 (2001), 832-851.

    This article has been revised from its preprint version which is available as a pdf file via the overview web-page WIAS-Preprint-552.

  24. N. Bubner, O. Klein, P. Philip, J. Sprekels, K. Wilmanski: A transient model for the sublimation growth of silicon carbide single crystals, Journal of Crystal Growth, 205 (1999), 294-304

    This article has been slightly revised from its preprint version which is available as a postscript or pdf file via the overview web-page WIAS-Preprint-443.

Articles in Conference Issues of Refereed Scientific Journals

  1. J. Geiser, O. Klein, P. Philip: Numerical simulation of temperature fields during the sublimation growth of SiC single crystals, using WIAS-HiTNIHS, Journal of Crystal Growth, 303 (2007), 352-356. (Proceedings of the 5th International Workshop on "Modeling in Crystal Growth" (IWMCG-5), 10-13/09/06, Bamberg, Germany.)

Articles in Conference Proceedings and Collections

  1. C. Meyer: An SQP active set method for a semilinear optimal control problem with nonlocal radiation interface conditions, In K. Kunisch, G. Leugering, J. Sprekels and F. Tröltzsch, editors, Control of Coupled Partial Differential Equations, Birkhäuser, Basel, 2007, 217,248.

  2. C. Meyer, I. Yousept: State-constrained optimal control problem with nonlocal radiation boundary conditions, Proc. Appl. Math. Mech., 2007. http://dx.doi.org/10.1002/pamm200700248

  3. J. Geiser, O. Klein, P. Philip: WIAS-HiTNIHS: A software tool for simulation in sublimation growth of SiC single crystals: Applications and Methods, to appear in the Proceedings of International Congress of Nanotechnology and Nano, 2004.

  4. E. Casas, F. Tröltzsch: , pp. 89-100 in V. Barbu, et al. (Eds): Analysis and Optimization of Differential Systems, Kluwer Academic Publishers, Boston, USA, 2003.

  5. J. Sprekels, O. Klein, P. Philip, K. Wilmanski: Optimal Control of Sublimation Growth of SiC Crystals, pp. 343-354 in W. Jäger, H.-J. Krebs (Eds): Mathematics - Key Technology for the Future, Springer-Verlag, Heidelberg, Germany, 2003.

  6. O. Klein, P. Philip: Induction Heating During SiC Growth by PVT: Aspects of Axisymmetric Sinusoidal Modeling, pp. 271-276 in Proceedings of the 10-th International IGTE Symposium on Numerical Field Calculation in Electrical Engineering, Graz, Austria, September 16-18, 2002, CD-ROM, ISBN: 3-901351-65-5. Using the following link, you can download the article in pdf-format.

  7. O. Klein, P. Philip: Transient Numerical Simulation of Sublimation Growth of SiC Single Crystals, pp. 127-136 in K.-H. Hoffmann (ed.): Smart Materials, Proceedings of the 1st caesarium, Bonn, November 17-19, 1999, Springer-Verlag, Berlin, Heidelberg, 2001.

Miscellaneous Articles

  1. O. Klein, P. Philip, J. Schefter, J. Sprekels: Numerical Simulation and Control of Sublimation Growth of SiC Bulk Single Crystals, in Annual Research Report 2002 of the Weierstrass Institute for Applied Analysis and Stochastics (WIAS), Berlin, Germany.

  2. W. Römisch, F. Tröltzsch: Applied nonlinear optimization (Section 3.1), pp. 21-37 in DFG Research Center Matheon "Mathematics for Key Technologies" publication of the Berlin Mathematical Society (page in German only) on the occasion of the opening ceremony of the DFG Research Center Matheon "Mathematics for Key Technologies", Berlin, Germany, November 2002.

  3. P. Philip: Produktion bei 3000 Grad mit kühlem Kopf optimieren (in German). Article provided for a press conference on the occasion of the opening ceremony of the DFG Research Center Matheon "Mathematics for Key Technologies", Berlin, Germany, November 2002. A pdf-version of this article is available from the press site of the Technical University (TU) Berlin, Germany. For convenience, you can you can also use this shortcut to the pdf-version.

  4. O. Klein, P. Philip, J. Sprekels, K. Wilmanski: Numerical simulation and optimization of SiC single crystal growth by sublimation from the gas phase, in Annual Research Report 2001 of the Weierstrass Institute for Applied Analysis and Stochastics (WIAS), Berlin, Germany.

  5. O. Klein, P. Philip, J. Sprekels, K. Wilmanski: Optimale Steuerung der Sublimationszüchtung von SiC-Kristallen (article and page in German), in Annual Research Report 2000 (report and page in German) of the Weierstrass Institute for Applied Analysis and Stochastics (WIAS), Berlin, Germany.

  6. O. Klein, P. Philip, J. Sprekels, K. Wilmanski: Optimale Steuerung der Sublimationszüchtung von SiC-Kristallen (article and page in German), in Annual Research Report 1999 (report and page in German) of the Weierstrass Institute for Applied Analysis and Stochastics (WIAS), Berlin, Germany.

  7. N. Bubner, O. Klein, P. Philip, J. Sprekels, K. Wilmanski: Optimale Steuerung der Sublimationsz´┐Żchtung von SiC-Kristallen (article and page in German), in Annual Research Report 1998 (report and page in German) of the Weierstrass Institute for Applied Analysis and Stochastics (WIAS), Berlin, Germany.

Presentations

The following list restricts to the presentations on sublimation growth. The presentations about the new topic of Project C9, crystal growth from the melt, are listed on the main page of the project.

International Conference Plenary Lectures

  1. J. Sprekels: Modeling and simulation of the sublimation growth of SiC bulk single crystals, Workshop "Interphase 2003: Numerical Methods for Free Boundary Problems", Newton Institute, Cambridge, UK, April 15, 2003.

    Use the following link to view the slides of the lecture.

International Conference Talks

  1. P. Philip: Optimal Control of Conductive-Radiative Temperature Fields Generated via Electromagnetic Heating, Conference on Optimal Control of Coupled Systems of PDE, Oberwolfach, Germany, 03/03/2008.

  2. F. Tröltzsch: Some results on state-constrained control problems for PDEs with finite-dimensional control space, MOSOCOP 2008, Heidelberg, Germany, 21/07/2008.

  3. F. Tröltzsch: Error estimates for elliptic control problems with finitely many constraints, ECCOMAS, Venice, Italy, 02/07/2008.

  4. F. Tröltzsch: Regularity and optimality conditions for state-constrained parabolic control problems with time-dependent controls, ENUMATH, Graz, Austria, 10/09/2007.

  5. F. Tröltzsch: Regularity of Lagrange multipliers for optimal control problems with PDEs and mixed control state constraints, 23rd TC7 Conference on System Modelling and Optimization, Cracow, Poland, 26/07/2007.

  6. F. Tröltzsch: A regularization method for elliptic boundary control problems with pointwise state constraints, 6th ICIAM, Zürich, Switzerland, 17.07.2007.

  7. F. Tröltzsch: From bottleneck problems to mixed control state contraints -- some results on multiplier regularity, RICAM-LINZ Workshop DFG-SPP 1253, Linz, Austria, 03.2007.

  8. I. Yousept: Optimal control of temperature distribution in seeded sublimation growth processes of semiconductor single crystal, ICIAM Minisymposium -- PDE-constrained optimization: numerical analysis and scientific computing, Zürich, Switzerland, 17.07.2007.

  9. I. Yousept: Semismooth Newton methods for Lavrentiev regularized nonlinear optimal control problems, DMV Minisymposium 2 - Numerics for PDE-Constrained Control Problems, Bonn, Germany, 21.09.2006.

  10. J. Geiser: WIAS-HiTNIHS: Software-tool for simulation in crystal growth for SiC single crystal : Application and Methods, The International Congress of Nanotechnology and Nano, San Francisco, USA, November 7-10, 2004.

  11. A. Rösch: Parabolic control problems with pointwise mixed control-state constraints, 21st IFIP TC 7 conference on System Modeling and Optimization, Sophia Antipolis, France, July, 2003.

  12. A. Rösch: Lavrentiev type regularization of optimal control problems with pointwise state constraints, international workshop on "Numerical and Symbolic Scientific Computing", St. Wolfgang / Strobl, Austria, June 18, 2003.

  13. F. Tröltzsch: Optimal control problems with mixed control-state constraints - duality regular Lagrage multipliers, conference on "Control of partial differential equations contstraints", USA, June, 2003.

  14. F. Tröltzsch: Numerical Analysis of Optimal Control Problems for PDEs, conference on "Control of partial differential equations", Spain, June, 2003.

  15. P. Philip: Towards Optimal Control of Sublimation Growth of SiC Bulk Single Crystals, minisymposium "Inverse Problems in Material Science" at the conference "Applied Inverse Problems: Theoretical and Computational Aspects", Lake Arrowhead, CA, USA, May 20, 2003.

    Using the following link, you can view and download the talk in pdf-format: talk.pdf.

  16. A. Rösch: Optimal error estimates for abstract linear-quadratic optimal control problems, conference on "Numerical Techniques for Optimization Problems with PDE Constraints", Oberwolfach, Germany, February 17, 2003.

  17. F. Tröltzsch: On error estimates for elliptic optimal control problems, conference on "Numerical Techniques for Optimization Problems with PDE Constraints", Oberwolfach, Germany, February 16, 2003.

  18. F. Tröltzsch: Optimal control of PDEs with mixed control-state constraints, 18th Int. Symposium on Math. Programming, Denmark, 2003.

  19. O. Klein: Surface movements during the sublimation growth of silicon carbide single crystals, workshop "Interfaces and Singularity Perturbed Interface Evolutions", Max-Planck-Institut für Mathematik in den Naturwissenschaften, Leipzig, Germany, November 29, 2002.

  20. F. Tröltzsch: A method of instantaneous control type for the heat equation, Gamm Workshop "Control Theory / Dynamical Systems", Berlin, Germany, November 2002.

  21. F. Tröltzsch: Error estimates for the optimal control of elliptic PDEs, CISC 2002, Berlin, Germany, October 2002.

  22. A. Rösch: Second-order conditions for parabolic control problems with pointwise mixed control-state constraints, French-German-Polish conference on Optimization, Cottbus, Germany, September 2002.

  23. F. Tröltzsch: On error estimates for the optimal control of PDEs, conference on "Analysis and Optimization of Differential Systems", Constanta, Romania, September 2002.

  24. P. Philip: Transient Numerical Simulation of Sublimation Growth of SiC Single Crystals, meeting on "Optimal Control of Complex Dynamic Structures", Oberwolfach, Germany, June 05, 2000.

  25. P. Philip: Transient Numerical Simulation of Sublimation Growth of SiC Single Crystals, PhTIEE "Phase Transitions and Interfaces in Evolution Equations: analysis, control, and approximation", Santa Margherita Ligure, Italy, February 09, 2000.

  26. P. Philip: Transient Numerical Simulation of Sublimation Growth of SiC Single Crystals, 1st caesarium "Smart Materials", caesar, Bonn, Germany, November 19, 1999.

  27. P. Philip: A Model for the Sublimation Growth of Silicon Carbide Single Crystals Including Heat Transfer by Radiation, workshop "Systems with Hysteresis", WIAS, Berlin, Germany, September 20, 1999.

  28. N. Bubner: On sublimation growth of silicon carbide single crystals, workshop "Phase Field Models and Surface Effects", Cortona, Italy, September 14, 1998.

Summer School Lecture

  • P. Philip: Stationäre und instationäre Simulation 3, WE-HERAEUS-FERIENKURS on "SiC and GaN - Materials for Power- and Opto-Electronics", Cottbus, Germany, September 03 - September 14, 2001.

Lecture to the General Public

  • O. Klein: Numerische Simulation und Optimierung der Züchtung von Halbleiter-Kristallen, Talk at the MATHEredaktiON talk series for school students of the DFG research center Matheon, WIAS Berlin, 18.09.2007.

  • D. Siche and P. Philip: Siliciumcarbid - ein Material mit Tradition und Zukunft (Silicon Carbide - a Material with Tradition and Future), Urania Berlin, Germany, January 24, 2002.

    The slides of P. Philip's part (Simulation) of the lecture (in German) are available in the Microsoft Powerpoint format (.ppt).

Talks at Colloquiums, Workshops, and Seminars

  1. F. Tröltzsch: Some results on restricted controllability, Chemnitz symposium on inverse problems, Germany, 25.09.2008.

  2. P. Philip: Numerical analysis and control of conductive-radiative heat transfer, Seminar talk at the Aachen Institute for Advanced Study in Computational Engineering Science (AICES), RWTH Aachen University, Germany, 29.01.2008.

  3. I. Yousept: State constrained optimal control of semilinear elliptic equations with nonlocal radiation interface conditions, Jour fixe of GK 1128, Berlin, Germany 24.01.2008.

  4. P. Philip: Numerics and optimal control of nonlinear conductive-radiative heat transfer, Seminar talk at the Mathematics Applications Consortium for Science and Industry, University of Limerick, Ireland, 20.07.2007

  5. P. Philip: Numerik und optimale Steuerung nichtlinearer Wärmeleitung mit Strahlung, Seminar talk at the Department of Mathematics, LMU Munich, Germany, 07.07.2007.

  6. P-E. Druet: Schwache Lösung für ein Modell elektromagnetischen Heizens mit Wärmestrahlung, Seminar Optimale Steuerung, TU Berlin, Germany, 24.05.2007.

  7. F. Tröltzsch: Optimale Steuerung partieller Differentialgleichungen mit Zustandsbeschränkungen, Colloquium at the University of Bayreuth, Germany, 02.01.2007.

  8. I. Yousept: A Lavrentiev type regularization approach for the numerical treatment of some state-constrained boundary control problems, Karl-Franzens-Universität, Graz, Austria, 01.2007.

  9. J. Geiser: Discretization- and Optimization methods for a Parabolic-Equation and application on simulation in crystal growth. Talk in the seminar parabolic differential equations, Technical University of Athen, Greece, November 29, 2004.

  10. J. Geiser: Simulations in crystal growth for SiC single crystal: Numerical Methods and Applications, Talk in the CASC/ISCR Seminar, Center for Applied Scientific Computing (CASC), Lawrence-Livermoore National Lab (LLNL), Livermoore, California, USA, November 10, 2004.

  11. J. Geiser: Discretization methods for Parabolic-Equations based on Finite-Volume- and related methods and Applications in Fluid- and Gas-Mechanics. at the Graduierten Kolleg: Nichtlineare Differentialgleichungen, Univesity of Freiburg, October 28, 2004.

  12. P. Philip: Numerical Simulation and Control of Sublimation Growth of SiC Bulk Single Crystals: Modeling, Finite Volume Method, Analysis and Results. Talk in the seminar Applied Mathematics and Numerical Analysis in the School of Mathematics of the University of Minnesota: Minneapolis, MN, USA, September 16, 2004.

  13. P. Philip: Numerical Simulation and Control of Sublimation Growth of SiC Bulk Single Crystals: Modeling, Finite Volume Method, Analysis and Results. Talk for the "Show and Tell" at the Institute for Mathematics and its Applications (IMA), University of Minnesota: Minneapolis, MN, USA, September 15, 2004.

  14. P. Philip: Numerical Simulation and Control of Sublimation Growth of SiC Bulk Single Crystals: Modeling, Finite Volume Method, Analysis and Results. Seminar talk at Corning Inc. Corning, NY, USA, September 08, 2004.

  15. O. Klein: Optimierung des Temperaturfeldes bei der Sublimationszüchtung von SiC Einkristallen, 3rd dgkk Workshop "Applied Simulation in Crystal Growth", Volkach am Main, Germany, Febuary 05, 2004.

  16. D. Schulz: Ein Benchmark zur SiC Kristallzüchtung, 3rd dgkk Workshop "Applied Simulation in Crystal Growth", Volkach am Main, Germany, Febuary 05, 2004.

  17. C. Meyer: Optimal control of sublimation growth of SiC bulk single crystals, talk at the meeting of the Application Area C of the DFG Research Center Matheon "Mathematics for Key Technologies", TU Berlin, Germany, Febuary 02, 2004.

  18. C. Meyer: Regularisierung von zustandsbeschränkten Optimalstuerproblemen, annual conference of the DMV, Rostock, Germany, September 2003.

  19. A. Rösch: Superkonvergenzeffekte bei Optimalsteueraufgaben, annual conference of the DMV, Rostock, Germany, September 2003.

  20. A. Rösch: Error estimates for constrained least squares problems, seminar talk, Linz, Austria, April 28, 2003.

  21. P. Philip: Defense Colloquium for the doctoral thesis Transient Numerical Simulation of Sublimation Growth of {SiC} Bulk Single Crystals. Modeling, Finite Volume Method, Results. Department of Mathematics, Humboldt University of Berlin, Germany, Febuary 03, 2003.

  22. F. Tröltzsch: Optimierung bei partiellen Differentialgleichungen - Beispiele und Methoden. Conference on "Mathematical Methods in the Economy, January, 2003.

  23. P. Philip: Produktion bei 3000 Grad mit kühlem Kopf optimieren, 4, opening ceremony of the DFG Research Center Matheon "Mathematics for Key Technologies", Berlin, Germany, November 20, 2002.

  24. A. Rösch: Approximation von Optimalsteueraufgaben durch endlichdimensionale Probleme, annual conference of the DMV, Halle, Germany, September 2002.

  25. D. Schulz, K. Böttcher, P. Philip: Erste Ergebnisse aus Simulationen mit Maxwell/FIDAP und WIAS-HiTNIHS: Vergleich mit Experiment, 5, Seminar of the research group Numerical Modelling of the Institute of Crystal Growth (IKZ), Berlin, Germany, October 29, 2002.

  26. P. Philip: WIAS-HiTNIHS - Transient Simulation with the High Temperature Numerical Induction Heating Simulator, 2nd dgkk Workshop "Applied Simulation in Crystal Growth", Memmelsdorf, Germany, October 10, 2002.

  27. F. Tröltzsch: Error estimates for the approximation of elliptic control problems. Conference on "Mathematical Methods in the Economy, Arnstadt, Germany, July, 2002.

  28. P. Philip: Numerical Simulation of SiC Bulk Single Crystal Growth by Sublimation, dgkk Workshop "Applied Simulation in Crystal Growth", Aufseß, Germany, Febuary 14, 2001.

  29. P. Philip: Zeitabhängige Modellierung der Sublimationszüchtung von SiC-Einkristallen 6, Colloquium of the Weierstrass Institute for Applied Analysis and Stochastics (WIAS), Berlin, Germany, April 17, 2000.

  30. P. Philip: Transient numerical simulation of sublimation growth of SiC single crystals, Weierstrass Institute for Applied Analysis and Stochastics (WIAS), Berlin, Germany, September 16, 1999.

  31. O. Klein: A model for the sublimation growth of silicon carbide single crystals including heat transfer by radiation, University of Pavia, Italy, September 05, 1999.

  32. J. Sprekels: Optimale Steuerung der Sublimations-Züchtung von SiC-Kristallen 7, BMBF-Statusseminar "Mathematische Verfahren zur Lösung von Problemstellungen in Industrie und Wirtschaft" 8, Erlangen, February 17 - February 19, 1999.

Poster Presentations

  1. O. Klein, P. Philip, J. Geiser, D. Siche and J. Wollweber: Numerical investigation of the influence of an anisotropic insulation felt on the temperature field in a PVT growth apparatus, 15th International Conference on Crystal Growth (ICCG15), Salt Lake City, USA, 12-17.08.2007.

  2. O. Klein, P. Philip, J. Geiser: Numerical simulation of temperature fields during the sublimation growth of SiC single crystals, using WIAS-HiTNIHS, 5th International Workshop on "Modeling in Crystal Growth" (IWMCG-5), Bamberg, Germany, 10-13.09.2006.

  3. O. Klein, J. Geiser, C. Meyer, P. Philip. J. Sprekels, F. Tröltzsch: Numerical Simulation and Control of Sublimation Growth of SiC Bulk Single Crystals, Workshop "Future Challenges in Multiscale Modeling and Simulation" at the Institute for Mathematics and its Applications (IMA), University of Minnesota, Minneapolis, MN, USA, November 18, 2004. You can use the following link to see a pdf slide version of this poster: slides.pdf.

  4. C. Meyer, P. Philip. F. Tröltzsch: Optimal Control of a PDE with Weakly Singular Integral Operator: Tuning Temperature Fields During Crystal Growth, Workshop "Singularities in Materials" at the Institute for Mathematics and its Applications (IMA), University of Minnesota, Minneapolis, MN, USA, October 25, 2004. You can use the following link to see a pdf slide version of this poster: slides.pdf.

  5. P. Philip: Poster for the "Show and Tell" at the Institute for Mathematics and its Applications (IMA), University of Minnesota: Numerical Simulation and Control of Sublimation Growth of SiC Bulk Single Crystals: Modeling, Finite Volume Method, Analysis and Results. Minneapolis, MN, USA, September 15, 2004.

  6. C. Meyer, O. Klein, P. Philip. A. Rösch, J. Sprekels, F. Tröltzsch: Optimalsteuerung bei der Herstellung von SiC-Einkristallen 9, public event MathInside - ueberall ist Mathematik 10 of the DFG Research Center Matheon "Mathematics for Key Technologies", Berlin, on the occasion of the Tag der Offenen Tür 11 at the Urania Berlin, September 13, 2003.

  7. O. Klein, P. Philip, J. Schefter, J. Sprekels: Numerical Simulation and Control of SiC Bulk Single Crystal Growth, BMBF-Statusseminar "Neue Mathematische Verfahren in Industrie und Dienstleistungen" 12, Ludwigshaven, Germany, December 16 - December 17, 2002. You can use the following link to see a .pdf version of this poster: poster.pdf

  8. O. Klein, P. Philip: Induction Heating During SiC Growth by PVT: Aspects of Axisymmetric Sinusoidal Modeling, 10-th International IGTE Symposium on Numerical Field Calculation in Electrical Engineering, Graz, Austria, September 16 - September 18, 2002.

  9. O. Klein: Modeling and Numerical Simulation of Induction Heating of Crucibles for Sublimation Growth of SiC, WE-HERAEUS-FERIENKURS on "SiC and GaN - Materials for Power- and Opto-Electronics", Cottbus, Germany, September 03 - September 14, 2001.

  10. O. Klein: Modeling and Numerical Simulation of Induction Heating of Crucibles for Sublimation Growth of SiC, ISIMM Conference "Continuum Mechanics & Thermodynamics", Potsdam, Germany, July 30 - August 3, 2001.

  11. O. Klein, P. Philip, J. Sprekels, K. Wilmanski: Numerical Simulation and Control of SiC Bulk Single Crystal Growth, BMBF-Statusseminar "Mathematische Verfahren zur Lösung von Problemstellungen in Industrie und Wirtschaft" and "Neue Mathematische Verfahren in Industrie und Dienstleistungen" 13, Frankfurt am Main, Germany, December 11 - December 12, 2000.

  12. N. Bubner, O. Klein, P. Philip, J. Sprekels, K. Wilmanski: A Transient Model for the Sublimation Growth of Silicon Carbide Single Crystals, Workshop "Crystal Growth Meeting Germany-Japan-Poland", Institute of Crystal Growth (IKZ), Berlin, Germany, April 19 - April 20, 1999.

  13. N. Bubner, O. Klein, P. Philip, J. Sprekels, K. Wilmanski: Optimale Steuerung der Sublimations-Züchtung von SiC-Kristallen 14, BMBF-Statusseminar "Mathematische Verfahren zur Lösung von Problemstellungen in Industrie und Wirtschaft" 15, Erlangen, February 17 - February 19, 1999.

  14. N. Bubner, O. Klein, P. Philip, J. Sprekels, K. Wilmanski: Optimale Steuerung der Sublimations-Züchtung von SiC-Kristallen 16, BMBF-Statusseminar "Mathematik - Schlüsseltechnologie für die Zukunft" 17, Heidelberg, Germany, 1997.



1 This work is a follow-up project of the one titled Optimal Control of Sublimation Growth of SiC crystals which has been supported by funding of the German Ministry for Education, Science, Research, and Technology under reference number 03SP7FV16 within the program "Mathematical Methods Solving Problems in Industry and Business"
2 Das diesem Bericht zugrundeliegende Vorhaben wurde bis Ende 2003 mit Mitteln des Bundesministeriums für Forschung und Technologie im Rahmen des Programms "Neue Mathematische Verfahren in Industrie und Dienstleistungen" unter Nr. 03SPM3B5 gefördert. Die Verantwortung für den Inhalt dieser Veröffentlichung liegt bei den Autoren
3 Stationary and transient simulation
4 Keeping Cool During Optimization of Production at 3000 Degrees
5 First Simulation Results Using Maxwell/FIDAP and WIAS-HiTNIHS: Comparison with Experiments
6 Transient Modeling of Sublimation Growth of SiC single crystals
7 Optimal Control of Sublimation Growth of SiC crystals
8 Meeting for projects supported by the German Ministry for Education, Science, Research, and Technology within the program "Mathematical Methods Solving Problems in Industry and Business"
9 Optimal Control During the Production of SiC Single Crystals
10 Everywhere, There Is Mathematics
11 Open Day
12 Meeting for projects supported by the German Ministry for Education, Science, Research, and Technology within the program "New Mathematical Methods in Manufacturing and Service Industry"
13 Meeting for projects supported by the German Ministry for Education, Science, Research, and Technology within the programs "Mathematical Methods Solving Problems in Industry and Business" and "New Mathematical Methods in Manufacturing and Service Industry"
14 Optimal Control of Sublimation Growth of SiC crystals
15 Meeting for projects supported by the German Ministry for Education, Science, Research, and Technology within the program "Mathematical Methods Solving Problems in Industry and Business"
16 Optimal Control of Sublimation Growth of SiC crystals
17 Meeting for projects supported by the German Ministry for Education, Science, Research, and Technology within the program "Mathematics - Key Technology for the Future"


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Last update: Olaf Klein on February 15th, 2012.